Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55

CAL2.122 Direct Comparison Test Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.122 Direct Comparison Test. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.122 Direct Comparison Test - Choose the correct benchmark and inequality to prove convergence or divergence. Standards tags: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.122 Direct Comparison Test skill assessments:

Problem 1Difficulty: Mastery

Which comparison proves convergence of \(\sum_{n=16}^\infty\frac{\ln(n+1)+1}{n²+1}\)?

ASince \(\ln(n+1)+1\le n\), compare above with \(1/n\) to prove convergence.

BSince \(a_n\ge1/n²\), comparison with \(\sum1/n²\) proves convergence.

CSince the numerator is unbounded, no \(p\)-series upper comparison is possible.

DFor \(n\ge16\), \(\ln(n+1)+1\le\sqrt n\) and \(n²+1\ge n²\), so \(0\le a_n\le1/n^3/2\), and the benchmark converges.

Solution & Explanation:The sharper eventual bound \(\ln(n+1)+1\le\sqrt n\) gives a convergent \(p=3/2\) upper comparison. The weaker upper bound \(1/n\) would be inconclusive.

Problem 2Difficulty: Mastery

For \(\sum_{n=1}^\infty\frac{1}{\sqrt{n²+n\sin n}}\), which comparison proves divergence?

ASince \(n²+n\sin n\ge n²-n\), compare above with \(1/\sqrt{n²-n}\) to prove convergence.

BSince \(a_n\ge1/n²\), comparison with \(\sum1/n²\) proves divergence.

CSince \(n²+n\sin n\le n²+n\le2n²\), \(\frac{1}{\sqrt{n²+n\sin n}}\ge\frac{1}{\sqrt2\,n}\), so the series diverges.

DSince a square root appears, the terms decrease fast enough for convergence.

Solution & Explanation:The denominator under the square root is at most \(2n²\), so the term is at least a constant multiple of \(1/n\). That harmonic lower bound proves divergence.

Problem 3Difficulty: Mastery

Which comparison proves \(\sum_{n=1}^\infty\frac{n²+∣\sin n∣}{n³+n∣\cos n∣}\) diverges?

ASince \(a_n\le2/n\), comparison with \(\sum2/n\) proves convergence.

BSince \(n²+∣\sin n∣\ge n²\) and \(n³+n∣\cos n∣\le2n³\), \(a_n\ge1/(2n)\), so the series diverges.

CSince \(a_n\ge1/n²\), comparison with \(\sum1/n²\) proves divergence.

DSince absolute values remove signs, the series automatically converges.

Solution & Explanation:Absolute values ensure nonnegative terms but do not force convergence. The terms dominate a harmonic multiple, so the series diverges.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55 (Choose the correct benchmark and inequality to prove convergence or divergence. Standards tags: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55).